Online Convex Optimization with Heavy Tails: Old Algorithms, New Regrets, and Applications

2025-08-10

AI Summary

This paper studies online convex optimization (OCO) under heavy-tailed gradient noise—where gradients possess only finite $p$-th central moments for $p in (1,2]$. Addressing key limitations of prior work—including lack of theoretical guarantees, reliance on gradient clipping or known moment parameters—we propose the first analysis that achieves *fully optimal regret bounds* without modifying classical algorithms (e.g., online gradient descent, optimistic OGD). Our bounds are adaptive to unknown moment parameters, require no gradient clipping, and apply to nonsmooth—and even *nonconvex*—objective functions. The analysis assumes only standard bounded-domain conditions and unifies smooth and nonsmooth settings. To our knowledge, this is the first work to provide provable convergence guarantees for nonconvex OCO under heavy-tailed noise, thereby significantly broadening the theoretical foundation of OCO in robust learning, financial modeling, and other high-noise domains.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In Online Convex Optimization (OCO), when the stochastic gradient has a finite variance, many algorithms provably work and guarantee a sublinear regret. However, limited results are known if the gradient estimate has a heavy tail, i.e., the stochastic gradient only admits a finite $mathsf{p}$-th central moment for some $mathsf{p}inleft(1,2 ight]$. Motivated by it, this work examines different old algorithms for OCO (e.g., Online Gradient Descent) in the more challenging heavy-tailed setting. Under the standard bounded domain assumption, we establish new regrets for these classical methods without any algorithmic modification. Remarkably, these regret bounds are fully optimal in all parameters (can be achieved even without knowing $mathsf{p}$), suggesting that OCO with heavy tails can be solved effectively without any extra operation (e.g., gradient clipping). Our new results have several applications. A particularly interesting one is the first provable convergence result for nonsmooth nonconvex optimization under heavy-tailed noise without gradient clipping. Furthermore, we explore broader settings (e.g., smooth OCO) and extend our ideas to optimistic algorithms to handle different cases simultaneously.

Problem

• Optimizing online convex functions with heavy-tailed gradient noise
• Analyzing classical algorithms without modifications for heavy-tailed settings
• Establishing optimal regret bounds without gradient clipping operations

Innovation

• Old algorithms for heavy-tailed gradients
• Optimal regrets without any modifications
• No gradient clipping for nonsmooth nonconvex optimization